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NAG Toolbox: nag_lapack_dpotri (f07fj)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dpotri (f07fj) computes the inverse of a real symmetric positive definite matrix A, where A has been factorized by nag_lapack_dpotrf (f07fd).

Syntax

[a, info] = f07fj(uplo, a, 'n', n)
[a, info] = nag_lapack_dpotri(uplo, a, 'n', n)

Description

nag_lapack_dpotri (f07fj) is used to compute the inverse of a real symmetric positive definite matrix A, the function must be preceded by a call to nag_lapack_dpotrf (f07fd), which computes the Cholesky factorization of A.
If uplo='U', A=UTU and A-1 is computed by first inverting U and then forming U-1U-T.
If uplo='L', A=LLT and A-1 is computed by first inverting L and then forming L-TL-1.

References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how A has been factorized.
uplo='U'
A=UTU, where U is upper triangular.
uplo='L'
A=LLT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The upper triangular matrix U if uplo='U' or the lower triangular matrix L if uplo='L', as returned by nag_lapack_dpotrf (f07fd).

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
U stores the upper triangle of A-1 if uplo='U'; L stores the lower triangle of A-1 if uplo='L'.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Diagonal element _ of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of A cannot be computed.

Accuracy

The computed inverse X satisfies
XA-I2cnεκ2A   and   AX-I2cnεκ2A ,  
where cn is a modest function of n, ε is the machine precision and κ2A is the condition number of A defined by
κ2A=A2A-12 .  

Further Comments

The total number of floating-point operations is approximately 23n3.
The complex analogue of this function is nag_lapack_zpotri (f07fw).

Example

This example computes the inverse of the matrix A, where
A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .  
Here A is symmetric positive definite and must first be factorized by nag_lapack_dpotrf (f07fd).
function f07fj_example


fprintf('f07fj example results\n\n');

% Lower triangular part of symmetric matrix A
uplo = 'Lower';
a = [ 4.16,  0,    0,    0;
     -3.12,  5.03, 0,    0;
      0.56, -0.83, 0.76, 0;
     -0.10,  1.18, 0.34, 1.18];

% Factorize A
[L, info] = f07fd( ...
                   uplo, a);

% Invert A using L.

[ainv, info] = f07fj( ...
                      uplo, L);

[ifail] = x04ca( ...
                 uplo, 'N', ainv, 'Inverse');


f07fj example results

 Inverse
             1          2          3          4
 1      0.6995
 2      0.7769     1.4239
 3      0.7508     1.8255     4.0688
 4     -0.9340    -1.8841    -2.9342     3.4978

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
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